I don’t know if EC Patriarchs outrank Cardinals of the Roman Church. Many EC’s would say that it is only appropriate that this should be so, but I doubt if Rome would agree. The fact that this topic is raised at the Middle Eastern Synod suggests that Cardinals officially, from Rome’s POV, do outrank EC Patriarchs (and there are Latin Patriarchs as well).
The role of the Cardinal is an exclusive Roman Church affair. His function is mainly to elect a pope. Other than ensuring that the EC churches have a say in the election of a pope, there is no reason why EC Patriarchs should be made Cardinals.
When the UGCC Patriarch Lubomyr received the Cardinal’s ring, he quickly took it off and put it in his pocket . . . he said he would only wear it when he would be in Rome.
Interestingly, a full 25% of Orthodox in Kyiv thought the primate of the UGCC is a very influential churchman. In my experience, Orthodox from Ukraine find it much more impressive that he is a Cardinal (I guess one too many patriarchs there already!). For a number of Ukrainian Orthodox, the Cardinalate is what links not only the UGCC but Ukraine itself with Europe etc. That is a separate discussion.
There is the argument that Eastern Catholics need not worry about participating in the election of a Pope since the Patriarch/other Primate is the immediate jurisdictional head of the EC Particular Churches which are in communion with the Roman Pontiff.
I think it is only proper for an EC Patriarch to be a Cardinal as a way to underscore the right of the EC Particular Churches to govern themselves, as elucidated by Vatican II, even from the vantage point of having a say in the election of a Pope.
The issue of precedence has a long and sometimes sad history with respect to the Eastern Catholic Churches.
For example, in RC Poland in the 17th and 18th centuries, during processions of the Catholic hierarchy, the Eastern Catholic Metropolitan-Archbishop of Kyiv followed only after the last Latin Catholic bishop. There was the view that Eastern Catholics, insofar as they followed the Eastern liturgical traditions seen as “schismatic” by Latin Catholics, were not “completely Catholic” as this would have entailed a complete adoption of the Latin Rite. There was the view that the Latin Rite was “higher” than any other Rite etc. and I have a Slavonic EC prayerbook that contains rules under which Latin Catholics could receive Holy Communion in an EC church - the rules are such that such “intercommunion” is entirely dissuaded for Latin Catholics. And so forth . . .
I think also that it is more to the point that Patriarchs work to restore all their rights as true “Orthodox Catholic Churches in communion with Rome” in the first instance.